/usr/share/doc/pyxplot/html/ex-eqnsolve.html is in pyxplot-doc 0.8.4-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 | <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
<head>
<meta name="generator" content="plasTeX" />
<meta content="text/html; charset=utf-8" http-equiv="content-type" />
<title>PyXPlot Users' Guide: Searching for Minima and Maxima of Functions</title>
<link href="sec-time_series.html" title="Working with Time-Series Data" rel="next" />
<link href="sect0024.html" title="Solving Systems of Equations" rel="prev" />
<link href="sect0019.html" title="Performing Calculations" rel="up" />
<link rel="stylesheet" href="styles/styles.css" />
</head>
<body>
<div class="navigation">
<table cellspacing="2" cellpadding="0" width="100%">
<tr>
<td><a href="sect0024.html" title="Solving Systems of Equations"><img alt="Previous: Solving Systems of Equations" border="0" src="icons/previous.gif" width="32" height="32" /></a></td>
<td><a href="sect0019.html" title="Performing Calculations"><img alt="Up: Performing Calculations" border="0" src="icons/up.gif" width="32" height="32" /></a></td>
<td><a href="sec-time_series.html" title="Working with Time-Series Data"><img alt="Next: Working with Time-Series Data" border="0" src="icons/next.gif" width="32" height="32" /></a></td>
<td class="navtitle" align="center">PyXPlot Users' Guide</td>
<td><a href="index.html" title="Table of Contents"><img border="0" alt="" src="icons/contents.gif" width="32" height="32" /></a></td>
<td><a href="sect0255.html" title="Index"><img border="0" alt="" src="icons/index.gif" width="32" height="32" /></a></td>
<td><img border="0" alt="" src="icons/blank.gif" width="32" height="32" /></td>
</tr>
</table>
</div>
<div class="breadcrumbs">
<span>
<span>
<a href="index.html">PyXPlot Users' Guide</a> <b>:</b>
</span>
</span><span>
<span>
<a href="sect0001.html">Introduction to PyXPlot</a> <b>:</b>
</span>
</span><span>
<span>
<a href="sect0019.html">Performing Calculations</a> <b>:</b>
</span>
</span><span>
<span>
<b class="current">Searching for Minima and Maxima of Functions</b>
</span>
</span>
<hr />
</div>
<div><h1 id="ex:eqnsolve">4.10 Searching for Minima and Maxima of Functions</h1>
<p>The <a name="a0000000475" id="a0000000475"></a><a name="a0000000476" id="a0000000476"></a> <tt class="tt">minimise</tt> and <tt class="tt">maximise</tt> commands can be used to find the minima or maxima of algebraic expressions. In each case, a single algebraic expression should be supplied for optimisation, together with a comma-separated list of the variables with respect to which it should be optimised. In the following example, a minimum of the sinusoidal function <img src="images/img-0010.png" alt="$\cos (x)$" style="vertical-align:-4px;
width:47px;
height:18px" class="math gen" /> is sought: </p><p> <tt class="tt">pyxplot> <b class="bf">set numerics real</b></tt><br /><tt class="tt">pyxplot> <b class="bf">x=0.1</b></tt><br /><tt class="tt">pyxplot> <b class="bf">minimise cos(x) via x</b></tt><br /><tt class="tt">pyxplot> <b class="bf">print x/pi</b></tt><br /><tt class="tt">1</tt> </p><p>Note that this particular example doesn’t work when complex arithmetic is enabled, since <img src="images/img-0010.png" alt="$\cos (x)$" style="vertical-align:-4px;
width:47px;
height:18px" class="math gen" /> diverges to <img src="images/img-0140.png" alt="$-\infty $" style="vertical-align:-1px;
width:30px;
height:9px" class="math gen" /> at <img src="images/img-0141.png" alt="$x=\pi +\infty i$" style="vertical-align:-1px;
width:92px;
height:13px" class="math gen" />. </p><p>Various caveats apply both to the <tt class="tt">minimise</tt> and <tt class="tt">maximise</tt> commands, as well as to the <tt class="tt">solve</tt> command. All of these commands operate by searching numerically for optimal sets of input parameters to meet the criteria set by the user. As with all numerical algorithms, there is no guarantee that the <i class="it">locally</i> optimum solutions returned are the <i class="it">globally</i> optimum solutions. It is always advisable to double-check that the answers returned agree with common sense. </p><p>These commands can often find solutions to equations when these solutions are either very large or very small, but they usually work best when the solution they are looking for is roughly of order unity. PyXPlot does have mechanisms which attempt to correct cases where the supplied initial guess turns out to be many orders of magnitude different from the true solution, but it cannot be guaranteed not to wildly overshoot and produce unexpected results in such cases. To reiterate, it is always advisable to double-check that the answers returned agree with common sense. </p><p> <span class="upshape"><span class="mdseries"><span class="rm">Finding the maximum of a blackbody curve.</span></span></span></p><div>
<table cellspacing="0" class="tabular">
<tr>
<td style="border-top-style:solid; border-left:1px solid black; border-right:1px solid black; border-top-color:black; border-top-width:1px; text-align:left"><p> When a surface is heated to any given temperature <img src="images/img-0142.png" alt="$T$" style="vertical-align:0px;
width:13px;
height:12px" class="math gen" />, it radiates thermally. The amount of electromagnetic radiation emitted at any particular frequency, per unit area of surface, per unit frequency of light, is given by the Planck Law: </p><table id="a0000000477" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="images/img-0143.png" alt="\[ B_\nu (\nu ,T)=\left(\frac{2h^3}{c^2}\right)\frac{\nu ^3}{\exp (h\nu /kT)-1} \]" style="width:281px;
height:45px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> The visible surface of the Sun has a temperature of approximately <img src="images/img-0144.png" alt="$5800\, \mathrm{K}$" style="vertical-align:0px;
width:52px;
height:13px" class="math gen" /> and radiates in such a fashion. In this example, we use the <tt class="tt">solve</tt>, <tt class="tt">minimise</tt> and <tt class="tt">maximise</tt> commands to locate the frequency of light at which it emits the most energy per unit frequency interval. This task is simplified as PyXPlot has a system-defined mathematical function <tt class="tt">Bv(nu,T)</tt> which evaluates the expression given above. </p></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p>Below, a plot is shown of the Planck Law for <img src="images/img-0145.png" alt="$T=5800\, \mathrm{K}$" style="vertical-align:0px;
width:90px;
height:13px" class="math gen" /> to aid in visualising the solution to this problem: </p></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><center>
<img src="images/img-0147.png" alt="\includegraphics[width=\textwidth ]{examples/eps/ex_eqnsolve}" style="width:" /> </center></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p>To search for the maximum of this function using the <tt class="tt">maximise</tt> command<a name="a0000000478" id="a0000000478"></a>, we must provide an initial guess to indicate that the answer sought should have units of Hz: </p></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p><tt class="tt">pyxplot> <b class="bf">nu = 1e14*unit(Hz)</b></tt><br /><tt class="tt">pyxplot> <b class="bf">maximise Bv(nu,5800*unit(K)) via nu</b></tt><br /><tt class="tt">pyxplot> <b class="bf">print nu</b></tt><br /><tt class="tt">340.9781 THz</tt> </p></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p>This maximum could also be sought be searching for turning points in the function <img src="images/img-0149.png" alt="$B_\nu (\nu ,T)$" style="vertical-align:-4px;
width:65px;
height:18px" class="math gen" />, i.e. by solving the equation </p><table id="a0000000479" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="images/img-0150.png" alt="\[ \frac{\mathrm{d}B_\nu (\nu ,T)}{\mathrm{d}\nu }=0. \]" style="width:116px;
height:39px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> This can be done as follows: </p></td>
</tr><tr>
<td style="text-align:left; border-right:1px solid black; border-left:1px solid black"><p><tt class="tt">pyxplot> <b class="bf">nu = 1e14*unit(Hz)</b></tt><br /><tt class="tt">pyxplot> <b class="bf">solve diff_dnu(Bv(nu,5800*unit(K)),nu) = <img src="images/img-0006.png" alt="$\backslash $" style="vertical-align:-5px;
width:7px;
height:18px" class="math gen" /></b></tt><br /><tt class="tt">.......> <b class="bf"> unit(0*W/Hz**2/m**2) via nu</b></tt><br /><tt class="tt">pyxplot> <b class="bf">print nu</b></tt><br /><tt class="tt">340.9781 THz</tt> </p></td>
</tr><tr>
<td style="border-bottom-style:solid; border-bottom-width:1px; border-left:1px solid black; border-right:1px solid black; text-align:left; border-bottom-color:black"><p>Finally, this maximum could also be found using PyXPlot’s built-in function <tt class="tt">Bvmax(T)</tt>:<br /><tt class="tt">pyxplot> <b class="bf">print Bvmax(5800*unit(K))</b></tt><br /><tt class="tt">340.9781 THz</tt> </p></td>
</tr>
</table>
</div></div>
<div class="navigation">
<table cellspacing="2" cellpadding="0" width="100%">
<tr>
<td><a href="sect0024.html" title="Solving Systems of Equations"><img alt="Previous: Solving Systems of Equations" border="0" src="icons/previous.gif" width="32" height="32" /></a></td>
<td><a href="sect0019.html" title="Performing Calculations"><img alt="Up: Performing Calculations" border="0" src="icons/up.gif" width="32" height="32" /></a></td>
<td><a href="sec-time_series.html" title="Working with Time-Series Data"><img alt="Next: Working with Time-Series Data" border="0" src="icons/next.gif" width="32" height="32" /></a></td>
<td class="navtitle" align="center">PyXPlot Users' Guide</td>
<td><a href="index.html" title="Table of Contents"><img border="0" alt="" src="icons/contents.gif" width="32" height="32" /></a></td>
<td><a href="sect0255.html" title="Index"><img border="0" alt="" src="icons/index.gif" width="32" height="32" /></a></td>
<td><img border="0" alt="" src="icons/blank.gif" width="32" height="32" /></td>
</tr>
</table>
</div>
<script language="javascript" src="icons/imgadjust.js" type="text/javascript"></script>
</body>
</html>
|